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The mathematically proven winning strategy for 14 of the most popular games


From Risk to tic-tac-toe, popular games involve tons of strategic decisions, probability and math. So one happy consequence of being a data nerd is that you may have an advantage at something even non-data nerds understand: winning.

So how do you win (almost) every game in existence, do you ask? Here are 20 data visualizations that offer lots of insight into the most popular games in America, including chess, Connect Four, Monopoly, Pac-Man, "Wheel of Fortune" and much more.
Battleship | Chess 1 | Chess 2 | Chess 3 | A coin toss | Connect Four | Diplomacy | "Jeopardy!" | Monopoly 1 | Monopoly 2 | Monopoly 3 | Monopoly 4 | Pac-Man | "The Price is Right" | Rock-paper-scissors | Scrabble | Texas hold ’em | Tic-tac-toe | Winning in Vegas


To win a game of Battleship, you need to do two things: maximize your probability of getting a hit at every turn, and hope your opponent doesn't do the same.

The probability of getting a hit on a ship on any square for a random board configuration is much higher for the lighter squares in the middle of the board, argues Alex Alemi, a graduate student in physics at Cornell who has blogged about the game. The graphic above shows that the probability ranges from about a 20 percent chance of hitting an opponent's ship when you pick a spot in the center of the board to 8 percent in the corner.

The reason is simple: There are a lot more ways to lay down a ship in the center. For the carrier, for example, there are only two ways to lay it so that there is a hit in the corner. But in the center there are 10: five ways to lay it horizontally and five vertically.

Chess, Part 1: The best squares to use on a chess board, according to the masters

This graphic by Seth Kadish of Vizual Statistix shows where notable chess players have moved their pieces most frequently during their careers.

The players tend to use a chevron in the center of the board: two central squares supported from behind by the two squares outwardly diagonal from those. In a typical game, this would take the form of a pawn in the center supported by a knight. The central squares that are a knight’s move distant are also shaded in the maps, showing that players often move the knight to those squares.

The graphic clearly shows that the player with the white pieces has an advantage: The masters moved their pieces into the opponent’s half of the board much less frequently when playing with black.

Chess, Part 2: The most popular first chess moves through history

This chart by self-described data tinkerer Randy Olson shows the most popular first move for the player with white in games of chess in modern history, followed by the black side’s response. The graphic is based on an extensive data set of chess games going back to 1850.

The first moves are usually pawn moves, and the destination squares for the pawns are labeled on the chart. In the Indian Defense, Black’s first move is with the knight, labeled with an “N.” (Click here for a complete guide to notation for chess.)

Chess, Part 3: What are the chances of survival for each chess piece?

Programmer and chess player Oliver Brennan designed a program to calculate the probabilities of survival for each piece using data from 2.2 million master-level tournament games.

Kings have the highest survival rate, of course, because they can’t be taken. Rooks also tend to be hardy because they spend a lot of time at the back of the board and are generally more active in endgames.

The knights and central pawns have the lowest survival rates. Many popular openings involve d and e pawns undertaking suicide missions, which are sometimes counter-attacked with c pawns. The wing pawns have a higher survival rate, prompting one forum user to comment, in what generally seems like a great rule for warfare, “If you can’t be the king, be the little guy hiding in the corner.”

A coin toss

Everyone knows there's a 50-50 chance of getting heads or tails, right? Well, maybe not.

An extensive study by Persi Diaconis, Susan Holmes and Richard Montgomery of Stanford found that if a coin is tossed and caught, it has about a 51 percent chance of landing on the same face it was launched. If a coin is spun rather than tossed, it has a more than 50 percent chance of landing on the heavier side (which is heads on a U.S. quarter). In the video above by Numerberphile, Diaconis explains some of the math behind the idea.

If this is true, how should you "win" a coin toss? First, always be the chooser and/or the tosser when you can. Don't allow the same person to both toss and choose -- unless that person is you. If the coin is being tossed and you're the chooser, choose the side that is face down. The coin is more likely to land with the same side up, but most people will invert the flipped coin onto their other hand before they reveal it.

Connect Four

Did you know there is a way to win at Connect Four literally 100 percent of the time? Connect Four is what mathematicians call a "solved game," meaning you can play it perfectly every time, no matter what your opponent does. You will need to get the first move, but as long as you do so, you can always win within 41 moves. Here's a mathy explanation, or you can watch the YouTube video above.


This map by Seth Kadish shows adjacency in the game of Diplomacy. Adjacency can be both a good and a bad thing, Kadish says, allowing for more options to support your units but also more ways to be attacked. If you're a more conservative player, you would want to play one of the more defensible positions: Turkey or France, rather than Germany or Austria.


Your probability of getting on "Jeopardy!" is in itself pretty slim, but if you do make it on, look to statistics to help you find the Daily Double. Nathan Yau of Flowing Data created this cool graphic showing the probability that Jeopardy’s Daily Double was found in a given location for seasons 1 through 31.

As Yau writes, Jeopardy fans criticized contestant Arthur Chu for jumping around the board, rather than choosing clues top to bottom in the traditional way. But his strategy was a good one: Historical probability suggests the Daily Double is much more likely to be found toward the bottom of the board than the top.

Monopoly, Part 1: Everyone needs a place to crash after a night in jail

Monopoly players are sent to jail for all kinds of reasons. And that means that the properties about one roll of the dice outside of jail are visited most often. As Walter Hickey, formerly of Business Insider and now at FiveThirtyEight, writes, jail serves as a "sink" -- people are sucked from places on the rest of the board and emerge from jail. That gives the orange and red properties a critical significance, as this video of 500 simulated rapid-fire Monopoly games shows.

Monopoly, Part 2: If you're not buying these properties, you're doing it wrong

The jail insight means the orange and red properties are generally the best investments, but they aren't the only ones. The graphic above, by Walter Hickey for Business Insider, shows the probability of landing on each space on the board. Notice the higher probability for spaces between jail and the "go to jail" space, as well as the railroads, and the lower probability of Chance spaces.

Monopoly, Part 3: Return on investment

There's one more thing you need to understand to play Monopoly strategically: how many houses to buy and where to put them. Tim Darling, who blogs at the site, calculated the "breakeven time" -- the number of opponent rolls that you need to make back the money that you invest in different numbers of houses on different properties. The properties are listed vertically at the left. The longest breakeven times are marked in blue, while the shortest are marked in red.

Right away, you can see that buying a single house on Mediterranean or Baltic Avenue is one of the worst uses of your money. Generally, you get the best return on your investment when you buy three houses on a property, or all four of the railroads. Owning one or two railroads is not as good -- though it does prevent your opponents from owning all four.

For a more thorough exploration of the mathematics of Monopoly, see Hickey's presentation here.

Monopoly, Part 4: How to win the fastest game of Monopoly ever

Ever wonder what the speediest game of Monopoly would look like? Profdjm computed the shortest theoretically possible game of Monopoly, which they demonstrate above. With the right sequence of rolls, Chance and Community cards, the four-turn, nine-roll game can be completed in just 21 seconds.


This graphic by Seth Kadish of Vizual Statistix shows you where to hang out and not to hang out while moonlighting as Pac-Man. The darker areas are more dangerous, since they are farther from an intersection. The lower half of the board is also more difficult to clear, says Kadish, especially the bottom row where you can get trapped by ghosts on either side.

There are a few special spots on the board: In the squares outlined in blue, ghosts cannot turn upward to follow you when they are in scatter and chase modes – only when they are frightened. And in the infamous hiding spot outlined in green you can sit indefinitely, assuming the ghosts didn’t see you move there, Kadish says.

"The Price is Right"

If you ever get to play on "The Price is Right," would you rather go home empty handed -- or with a new car? To go home with all the prizes, you don't need to know the price of a karaoke machine, an oven range or Reese's Peanut Butter Cups -- all you need is to understand some strategy, explains Ben Blatt at Slate.

Blatt argues that people can use game theory -- a school of thought in economics that looks at how people make decisions -- to win lots of things on "The Price is Right." For example, in the show's first segment, Contestants' Row, four people are chosen from the audience to guess the price of an item. The contestant whose guess is closest to the actual price without overbidding wins the prize and continues on to more lucrative games. If you are the last contestant, game theory says that you should bid one dollar more than the highest bidder, Blatt says. Blatt looked at 1,500 Contestants' Row games and found that if final contestants had used this strategy, they would have won 54 percent of the time. Instead, they won only 35 percent of the time.

The graphic above shows his solution for another "Price is Right" game, called Now or Then, in which contestants have to guess whether each of six small prices is labeled with its current price, or a price from an earlier date. Four items are always Now and two are always Then. Blatt says that most contestants try to figure out which prices are earlier and which are later, but instead they should just follow the decision-tree above. Like Connect-Four, Now and Then is a solved game, meaning that if you use the right strategy you can win 100 percent of the time.


If you were playing the familiar game of rock-paper-scissors against a computer, probability dictates that you would do equally well by picking any one of the three options. Whether you pick paper, scissors or rock, your chance of winning would always be one in three. But because people are far less random than computers, you can devise a strategy that will give you a greater chance of winning when playing rock-paper-scissors against a human.

This video by Numberphile breaks down the best strategy for winning at rock-paper-scissors, drawn from an awesome study of a massive tournament at China’s Zhejiang University (The Post’s Caitlin Dewey explained that study here). Basically, when people win they tend to stick with the same choice of rock, paper or scissors, while when they lose they are much more likely to switch to another option.


Texas hold ’em

Chris Beaumont, a software engineer at Counsyl, created some cool visualizations showing the strength of different hands in Texas hold ’em. The graphs are based on Beaumont’s enumeration of all roughly 1.3 trillion hands of heads-up Texas hold ’em, plus data on several million online poker hands.

The graphic requires a little explanation: The one on the top left shows the strength of a hand averaged over all opponent hands, with blue indicating hands that win more often than lose, and pink squares hands that lose more often than win. This graphic doesn't take into account that opponents can fold weaker hands, meaning your hand won’t perform as well as this graph suggests. The graph on the top right shows that missing data point: the actual frequency with which each hand is played. The brightest squares indicate the hands that are played most often, and the darker squares indicate those that are usually folded.

The graph on the bottom left combines those two data sets, providing a more accurate way of looking at average hand strength. Each hand is averaged over all opponent hands, but the average is also weighted by the frequency of the opponent hand. Again, blue hands win more often than lose, while pink hands lose more often than win.

Finally, the graph on the bottom right shows the strength for one particular hand, here an eight of hearts and a queen of spades. The graph now represents the opponent’s hand; the redder the square, the more likely the opponent is to win. This is an interactive: You can visit Beaumont’s site to try out the odds for different hands.


Randall Munroe at web comic created this amazing map of the optimal moves for X’s and O’s.

The diagram looks complicated, but it's actually pretty simple: Use the top graph when you are X and the bottom graph when you are O. Your optimal move is given by the position of the largest red symbol on the grid. When your opponent picks a move, zoom in on that region of the grid where they moved and pretend as if that is the entire grid, again selecting the largest red symbol. Repeat until the game ends.

"Wheel of Fortune"

If you ever go on "Wheel of Fortune" and manage to make it to the final round, choose the letters G, H, P and O, says Wonkblog's Chris Ingraham. A quick refresher: The contestant who wins the most money during regular play on "Wheel of Fortune" gets to try a bonus puzzle at the end of the show. Vanna White automatically flips over the letters R, S, T, L, N and E, and then you get to choose three more consonants and a vowel before solving the puzzle.

Ingraham's analysis of 1,546 "Wheel of Fortune" episodes between 2007 and 2014 shows that different letters appear at much different frequencies in that bonus puzzle. You'll notice that all the letters that Vanna gives you (R, S, T...) are relatively under-represented in the puzzle. So are the letters that people choose most often -- C, D, M and A. It turns out that the three consonants and the vowel that will give you the best odds of winning are G, H, P and O -- compared with CDMA, you're twice as likely to get four or more letters revealed, and half as likely to get nothing.

Winning in Vegas

If you're going to Vegas expecting to win, here's a reality check: You probably won't. There are all kinds of strategies out there to win at poker, blackjack and other casino games, but the fact remains that the house is likely to take a significant percentage of your money.

If you want to minimize your losses, however, you can use the first graph above from Seth Kadish. The horizontal axis shows the total monetary value of bets placed in millions of dollars, while the vertical axis shows the percent of wagered money won by the house between March 1, 2013, and Feb. 28, 2014, at non-restricted locations in the Las Vegas Strip area.

If you want to lose a smaller percentage of your money, the data suggests you should focus on the games that appear toward the bottom of that first graph. Your best bet is the $100 slot machines, where casinos take only 3.6 percent of your money. If you aren't rolling in Benjamins, you can try the penny slots, where you lose only 11.8 percent.

If you're looking just at sports, betting on baseball will give you slightly better odds than basketball and football, and much better odds than racing. For table games, bingo is the clear winner, with the house taking on 8.8 percent of all wagers, followed by blackjack (11.1 percent). The worst table game is three-card poker, where the house takes almost a third of all wagers.